On the hydrodynamic instabilities in boundary-layer flows induced by rotating tori

Abstract

We present a linear instability analysis of three-dimensional laminar boundary-layer flows over the outer surfaces of rotating families of tori in a quiescent, incompressible fluid using toroidal-poloidal coordinates. The torus, defined by its aspect ratio (major radius to tube radius), includes spindle, horn, and ring tori. With azimuthal symmetry, the analysis is local in the poloidal angle, beginning at a pole or hub circle. The perturbation equations are reduced to sixth-order ordinary differential equations, solved as an eigenvalue problem by a fourth-order Runge-Kutta shooting method with the Gram-Schmidt orthonormalization. A Newton-Raphson procedure then satisfies the dispersion relation to determine spatial stability. Neutral curves for stationary vortices are computed for poloidal angles up to 70 degrees from a pole or hub using a Reynolds number based on the sphere's equatorial radius for aspect ratios from 0 to 7. For an aspect ratio of 0, we reproduce the convective instability results of the rotating sphere. For each fixed torus, higher rotation rates are needed near the pole or hub, while lower rates suffice near the equator. As the aspect ratio increases, lower rotation rates trigger instability at all angles. For aspect ratios below 0.15, crossflow modes dominate up to 60 degrees, with streamline-curvature modes prevailing beyond. For aspect ratios above 0.15, curvature effects vanish but reappear weakly below -30 degrees for aspect ratios above 1.5. Vortex counts and their variation with rotation rates are discussed for all families of tori. Finally, the assumption of non-stationary vortices is considered for the spindle tori.